1. 1 リーマン面の退化現象２ （これからのこと） －普遍退化族？の構成－ 松本幸夫（学習院大学理学部） 札幌幾何学セミナー 2009/02/17

2. 2 Notation : a compact Riemann surface of genus : mapping class group of : Teichmüller space of complex analytic space acts -analytically : moduli space = moduli space of stable curves = compactification of (“Deligne-Mumford compactification” 1969) ・ ・ ・ nodes

3. 3 Recall: Bers Theorem (Acta Math. 130. 1973 p103) fiber space s.t. over a point we have the fiber Compactification

4. 4 The aim of this talk is to prove Theorem The compactified fiber space is the universal degenerating family, i.e. for any fiber space (of genus ) with non-constant moduli: (at present ) “pull back diagram”

5. 5 Symmetry of Riemann surface at finite or infinite palce ideaBers : Acta Math. 141 (1978) Bers – Thurston classification of mapping classes 1.periodic (elliptic) 2.hyperbolic (“irreducible” not elliptic) (pseudo-Anosov) 3.parabolic “reduced” by periodic 4.pseudo-hyperbolic (“reducible” not parabolic) acts on : periodic has fixed points in : parabolic has fixed points at of

6. 6 periodic parabolic hyperbolic cf. fixed point

7. 7 idea (bis) Classification of degenerating family over topological monodromy Def : pseudo-periodic : periodic or : parabolic Theorem (M. & Montesinos 1991, 1992) Bull. AMS ’94 bijection pseudo-periodic mapping classes of negative twist conjugation : top. monodromy

8. 8 Bers constructed a fiber space (by simultaneous uniformization) Fuchsian group “Extended modular group” acts on : normalizer of “modular group” “tautological fiber space” acts on in : id. on

9. 9 : periodic ● quotient ● ● singular fiber whose topological monodromy = periodic

10. 10 : parabolic ● quotient ● “compactified” singular fiber whose topological monodromy = parablolic

11. 11 Some details (well-known) (geodesic) pants decomposition closed geodesics Fenchel – Nielsen Coordinates “geodesic length”“twisting angle”

12. 12 Two basic Lemmas (cf. Abikoff’s Lecture Note, LNM 820 pp 95- ) Lemma 1 a universal constant s.t. geodesic simple loops with “short simple closed curves do not intersect” Lemma 2 a universal constant s.t. every Riemann surface has a pants decomposition with curves of length long short

13. 13 Compactification Process of Given a set of infinite # of points By the action of, we may assume infinite # of points w.r.t. some pants decomposition (Fenchel – Nielsen coordinates) Thus either 1. convergent subsequence → or 2. (nodes)

14. 14 To describe the second case, Bers introduced “Deformation space” Riemann surface with nodes free abelian group generated by Dehn twists “completion” of

15. 15 (“off axis” part) ( )

16. 16 Allowable map (Bers) deformation allowable “infinite cyclic covering”

17. 17 To obtain, we must further make “quotient” of. But we cannot “see” the action of on, because the action of is not well-defined on. Def normalizer of in acts on biholomorphically. Teichmüller space of Riemann surfaces of nodes

18. 18 small -neighborhood of in where preserves In Bers classification: If is parabolic, reduced by, is periodic in If is pseudo-hyperbolic, reduced by, is hyperbolic in

19. 19 We can construct the compactification as an orbifold. Orbifold charts: and tautological fiber space (with smooth fiber) on but on, we have singular fiber with finite monodromy family of Riemann surfaces with nodes on but on, we have singular fiber with pseudo-periodic monodromy (cf. I. Kra 1990) Type 1 Type 2

20. 20 Example “parabolic” 180° gives a full Dehn twist here The corresponding singular fiber is 2

21. 21 fixed points ofquotient 2 MM Thm

22. 22 By Mat. – Montesinos Thm, there are no singular fibers other than the above two types. Thus the compactified fiber space over contains (essentially) all types of singular fibers over. The existence of the pull-back diagram in the Theorem could be proved by the method of Imayoshi (1981).

23. 23 Thank you!